1. ## spherical shell

Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, $\displaystyle \rho = kr$, for some constant $\displaystyle k$.

So $\displaystyle \oint \bold{E} \cdot d \bold{a} = \frac{Q_{\text{enc}}}{\epsilon_{0}}$.

And $\displaystyle E \cdot 4 \pi r^{2} = \frac{1}{\epsilon_{0}} \int \rho \ d \tau$.

So $\displaystyle E \cdot 4 \pi r^{2} = \frac{k}{\epsilon_{0}} \int_{0}^{2 \pi} \int_{-\pi/2}^{\pi/2} \int_{0}^{r} r'^{3} \sin \theta \ d r' d \theta \ d \phi$. Then solve for $\displaystyle \bold{E}$.

Is this correct?

2. Originally Posted by heathrowjohnny
Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, $\displaystyle \rho = kr$, for some constant $\displaystyle k$.

So $\displaystyle \oint \bold{E} \cdot d \bold{a} = \frac{Q_{\text{enc}}}{\epsilon_{0}}$.

And $\displaystyle E \cdot 4 \pi r^{2} = \frac{1}{\epsilon_{0}} \int \rho \ d \tau$.

So $\displaystyle E \cdot 4 \pi r^{2} = \frac{k}{\epsilon_{0}} \int_{0}^{2 \pi} \int_{-\pi/2}^{\pi/2} \int_{0}^{r} r'^{3} \sin \theta \ d r' d \theta \ d \phi$. Then solve for $\displaystyle \bold{E}$.

Is this correct?
Looks fine. (Although the symbols for azimuthal and polar angles in standard usage are the other way around. But your integral limits and order of integration is consistent with this so there's no problem).